A

2. Let f(x,y) = (x² + y²)^-1/2 for (x,y) not equal to zero. What is the directional derivative of f at the point (x,y) in the direction toward the origin?

A. 1	B. (1/2)[(x+y)/(x²+y²)^3/4]	C. 1/(x²+y²)^3/2
D. (1/2)[(x+y)/(x²+y²)]	E. 1/(x²+y²)

Solution: The answer is E

By a theorem, if f is a differentiable function of x and y, then f has a directional derivative in the direction of any unit vector u = <a,b>, defined as,

The partial derivative of f with respect to x is given by,

The partial derivative of f with respect to y is given by,

The unit vector u at the point (x,y) in the direction toward the origin is given by,

Thus, by a theorem, the directional derivative of f at the point (x,y) in the direction toward the origin is,